Preconditioned Krylov subspace methods are suitable only when the action of the preconditioner is cheap to apply. It is, therefore, of interest to apply the preconditioners in Algorithm REF , i.e., the inexact algorithm, in the most efficient way possible. Inspired by work in [1]}, we quantify how inexactly the preconditioner \((K-\sigma M)^{-1}\)  can be applied at each iteration of our algorithm when approximating the solution to the shifted linear system (), without degrading convergence in general. In particular, in this section we prove a bound on the quantity \(\delta _i\)  in (REF ), i.e., we specify under which circumstances the inexact algorithm applied to () converges, given that Algorithm , i.e., the exact algorithm, applied to the same system converges.
